where T is the temperature of the larger subsystem. Hence

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f 2 (E - E1 ) == exp

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(7.4)

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The first factor is independent of E 1 and is thus a constant as far as the small subsystem is concerned. Owing to (7.2) and the fact that E 1 = Jlt1(Pl, q1)' we may take the ensemble density for the small subsystem to be

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p(p, q) = e-.Yf'(p,q)/kT

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(7.5)

where the subscript 1 labeling the subsystem has been omitted, since we may now forget about the larger subsystem, apart from the information that its temperature is T. The larger subsystem in fact behaves like a heat reservoir in thermodynamics. The ensemble defined by (7.5), appropriate for a system whose temperature is determined through contact with a heat reservoir, is called the canonical ensemble, The volume in f space occupied by the canonical ensemble is called the partition function: d 3N d 3N p q e-{3.Yf'(p, q) (7.6) Q ( V T) == N' N!h3N

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where /3 = l/kT, and where we have introduced a constant h, of the dimension of momentum X distance, in order to make QN dimensionless. The factor liN! appears, in accordance with the rule of "correct Boltzmann counting." These constants are of no importance for the equation of state. Strictly speaking we should not integrate over the entire f space in (7.6), because (7.2) requires that P(P1' q1) vanish if E 1 > E. The justification for ignoring such a restriction is that in the integral (7.6) only one value of the energy JIt (p, q) contributes to the integral and that this value will lie in the range where the approximation (7.4) is valid. We prove this contention in Section 7.2. The thermodynamics of the system is to be obtained from the formula

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(7.7)

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where A(V, T) is the Helmholtz free energy. To justify this identification we show

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CANONICAL ENSEMBLE AND GRAND CANONICAL ENSEMBLE

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(a) A is an extensive quantity,

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(b) A is related to the internal energy U == (H) and the entropy S == A Th by the thermodynamic relation

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- (a ja

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That A is an extensive quantity follows from (7.6), because if the system is made up of two subsystems whose mutual interaction can be neglected, then QN is a product of two factors. To prove the relation (b), we first convert (b) into the following differential equation for A:

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A - T( aA) aT v

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(7.8)

To prove (7.8), note the identity

_ _ jdpdqe{3[A(V,T)-.Yi'(P,q)] N!h 3N

(7.9)

Differentiating with respect to

/3 on both sides, we obtain

.Yt(p, q)

N!~3N f dp dqe{3[A(V, T)-.Yi'(p, q)] [A(V, T) This is the same as

+ /3(

:; )

A(V, T) - U(V, T) - T( aA) = 0 aT v

All other thermodynamic functions may be found from A(V, T) by the Maxwell relations in thermodynamics:

(aA)

S = _ (aA )

aT v

G =A + PV U = (H) = A

+ TS

Therefore all calculations in the canonical ensembles begin (and nearly end) with the calculation of the partition function (7.6).

7.2 ENERGY FLUCTUATIONS IN THE CANONICAL ENSEMBLE We now show that the canonical ensemble is mathematically equivalent to the microcanonical ensemble in the sense that although the canonical ensemble contains systems of all energies the overwhelming majority of them have the same

STATISTICAL MECHANICS

energy. To do this we calculate the mean square fluctuation of energy in the canonical ensemble. The average energy is

j dp dq Jft'e- {3.Yf' u= (Jft') = j dpdqe-{3.Yf'

(7.10)

Hence

j dpdq [U - Jft'(p, q)] e{3[A(V,T)-.Yf'(p,q ) = 0

(7.11)

Differentiating both sides with respect to

we obtain

+ jdpdqe{3(A-.Yf') (U _ Jft')(A _ Jft'-

(7.12)

By (7.8) this can be rewritten in the form au a/3 + ((U - Jft')2)

(7.13)

Therefore the mean square fluctuation of energy is

(Jft'2) - (Jft')2

kT 2 v C

(7.14)

For a macroscopic system (Jft') 0:: Nand C v 0:: N. Hence (7.14) is a normal fluctuation. As N --+ 00, almost all systems in the ensemble have the energy (Jft'), which is the internal energy. Therefore the canonical ensemble is equivalent to the microcanonical ensemble. It is instructive to calculate the fluctuations in another way. We begin by calculating the partition function in the following manner:

_l_jdpdqe-{3.Yf'(p,q) = N!h 3N

1 dEw(E) e-{3E = 1 dEe-{3E+!ogw(E)

00 00

faoo dE e{3[TS(E)-E)

(7.15)

where S is the entropy defined in the microcanonical ensemble. Since both Sand U are proportional to N, the exponent in the last integrand is enormous. We expect that as N --+ 00 the integral receives contribution only from the neighborhood of the maximum of the integrand. The maximum of the integrand occurs at E = E, where E satisfies the conditions

(7.16)

(7.17)